Probably most students of mathematics are somewhat suspicious of complex
numbers when first introduced to them. They seem to be predicated upon the
existence of a logical contradiction; i.e., the square root of minus one.
Any positive number times itself is a positive number. Any negative number
times itself is likewise a positive number. Therefore a number times itself
that is negative seems impossible. And so it is impossible for any real
number. To hypothesize the existence of a square root of minus one and call
it imaginary does not assuage the skepticism of the student. But there is
a simpler, more concrete way to define the complex numbers.

Complex numbers are ordered pairs of real numbers for which multiplication
is defined in a special way. Let (a,b) and (c,d) be ordered pairs of real
numbers. The product of (a,b) with (c,d) is defined as:

(a.b)*(c,d) = (ac-bd,ad+bc).

For any two pairs such that the second components are zero:

(a,0)*(c,0) = (ac,0)

Thus such ordered pairs with second component zero behave exactly like real
numbers.

Consider now the product of the complex number (0,1) with itself. The
result is:

(0,1)*(0,1) = (0-1,0+0) = (-1,0).

So (0,1) is a number such that its square is equal to (-1,0) which is
equivalent to -1. Thus the square root of -1 is not fictitious or
imaginary in the ordinary sense but is simply (0,1).

The negative numbers can be considered to have arisen as solutions
of algebraic equations of the form x+a=0. The rational numbers can be
considered to have arisen as solutions of algebraic equations of the
form ax+b=0 for a and b integral. The irrational numbers can be considered
to have arisen as solutions of algebraic equations of the form
x²-2=0. Likewise complex numbers can be considered to have arisen
as solutions of algebraic equations of the form x²+1=0. The question
occurs as to whether new types of numbers arise in the solution of
equations of the form ax²+bx+c=0 if the coefficients a, b and c are
complex. The answer is no. Complex numbers are as far as we have to
go. There is a theorem, called the Fundamental Theorem of Algebra, that
establishes that any polynomial equation with complex coefficients will have
all its roots as complex numbers.

The addition of complex numbers is very simple; the sum of two complex
numbers is the complex numbers whose components are the sums of the
components of the two numbers. That is to say,

(a,b) + (c,d) = ((a+c),(b+d))

The representation of complex numbers given above is the rectangular
representation. There is another equivalently representation of complex
numbers that is more convenient for some purposes. It is the polar
representation of complex numbers. A complex number as a point in a
two dimensional plane can be completely characterized by a radius (the
distance of the point from the origin) and an angle. The radius of a
complex number is also called its modulus. The relationship
between the rectangular representation (x,y) or x+iy and the polar
representation (r,θ) is:

r = (x2+y2)1/2
θ = tan-1(y/x)

One of the virtues of the polar representation is that multiplication of
two complex numbers is simple; i.e.,

(r,θ)*(s,φ) =
(rs, θ+φ)

Thus finding the power of a complex number in polar form is especially
easy; i.e.,

(r,θ)n = (rn ,nθ)

For a web calculator of a complex number to a complex power click
here.